lax_scipy_sparse_test.py

# Copyright 2020 The JAX Authors.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     https://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.

from functools import partial
import unittest

from absl.testing import absltest
import numpy as np
import scipy.sparse.linalg

from jax import jit
import jax.numpy as jnp
from jax import lax
from jax._src import dtypes
from jax._src import test_util as jtu
from jax.tree_util import register_pytree_node_class
import jax.scipy.sparse.linalg
import jax._src.scipy.sparse.linalg as sp_linalg

from jax.config import config
config.parse_flags_with_absl()


float_types = jtu.dtypes.floating
complex_types = jtu.dtypes.complex


def matmul_high_precision(a, b):
  return jnp.matmul(a, b, precision=lax.Precision.HIGHEST)


@jit
def posify(matrix):
  return matmul_high_precision(matrix, matrix.T.conj())


def solver(func, A, b, M=None, atol=0.0, **kwargs):
  x, _ = func(A, b, atol=atol, M=M, **kwargs)
  return x


lax_cg = partial(solver, jax.scipy.sparse.linalg.cg)
lax_gmres = partial(solver, jax.scipy.sparse.linalg.gmres)
lax_bicgstab = partial(solver, jax.scipy.sparse.linalg.bicgstab)
scipy_cg = partial(solver, scipy.sparse.linalg.cg)
scipy_gmres = partial(solver, scipy.sparse.linalg.gmres)
scipy_bicgstab = partial(solver, scipy.sparse.linalg.bicgstab)


def rand_sym_pos_def(rng, shape, dtype):
  matrix = np.eye(N=shape[0], dtype=dtype) + rng(shape, dtype)
  return matrix @ matrix.T.conj()


class CustomOperator:
  def __init__(self, A):
    self.A = A
    self.shape = self.A.shape

  def __matmul__(self, x):
    return self.A @ x


class LaxBackedScipyTests(jtu.JaxTestCase):

  def _fetch_preconditioner(self, preconditioner, A, rng=None):
    """
    Returns one of various preconditioning matrices depending on the identifier
    `preconditioner' and the input matrix A whose inverse it supposedly
    approximates.
    """
    if preconditioner == 'identity':
      M = np.eye(A.shape[0], dtype=A.dtype)
    elif preconditioner == 'random':
      if rng is None:
        rng = jtu.rand_default(self.rng())
      M = np.linalg.inv(rand_sym_pos_def(rng, A.shape, A.dtype))
    elif preconditioner == 'exact':
      M = np.linalg.inv(A)
    else:
      M = None
    return M

  @jtu.sample_product(
    shape=[(4, 4), (7, 7)],
    dtype=[np.float64, np.complex128],
    preconditioner=[None, 'identity', 'exact', 'random'],
  )
  def test_cg_against_scipy(self, shape, dtype, preconditioner):
    if not config.x64_enabled:
      raise unittest.SkipTest("requires x64 mode")

    rng = jtu.rand_default(self.rng())
    A = rand_sym_pos_def(rng, shape, dtype)
    b = rng(shape[:1], dtype)
    M = self._fetch_preconditioner(preconditioner, A, rng=rng)

    def args_maker():
      return A, b

    self._CheckAgainstNumpy(
        partial(scipy_cg, M=M, maxiter=1),
        partial(lax_cg, M=M, maxiter=1),
        args_maker,
        tol=1e-12)

    self._CheckAgainstNumpy(
        partial(scipy_cg, M=M, maxiter=3),
        partial(lax_cg, M=M, maxiter=3),
        args_maker,
        tol=1e-12)

    self._CheckAgainstNumpy(
        np.linalg.solve,
        partial(lax_cg, M=M, atol=1e-10),
        args_maker,
        tol=1e-6)

  @jtu.sample_product(
    shape=[(2, 2)],
    dtype=float_types + complex_types,
  )
  def test_cg_as_solve(self, shape, dtype):

    rng = jtu.rand_default(self.rng())
    a = rng(shape, dtype)
    b = rng(shape[:1], dtype)

    expected = np.linalg.solve(posify(a), b)
    actual = lax_cg(posify(a), b)
    self.assertAllClose(expected, actual, atol=1e-5, rtol=1e-5)

    actual = jit(lax_cg)(posify(a), b)
    self.assertAllClose(expected, actual, atol=1e-5, rtol=1e-5)

    # numerical gradients are only well defined if ``a`` is guaranteed to be
    # positive definite.
    jtu.check_grads(
        lambda x, y: lax_cg(posify(x), y),
        (a, b), order=2, rtol=2e-1)

  def test_cg_ndarray(self):
    A = lambda x: 2 * x
    b = jnp.arange(9.0).reshape((3, 3))
    expected = b / 2
    actual, _ = jax.scipy.sparse.linalg.cg(A, b)
    self.assertAllClose(expected, actual)

  def test_cg_pytree(self):
    A = lambda x: {"a": x["a"] + 0.5 * x["b"], "b": 0.5 * x["a"] + x["b"]}
    b = {"a": 1.0, "b": -4.0}
    expected = {"a": 4.0, "b": -6.0}
    actual, _ = jax.scipy.sparse.linalg.cg(A, b)
    self.assertEqual(expected.keys(), actual.keys())
    self.assertAlmostEqual(expected["a"], actual["a"], places=6)
    self.assertAlmostEqual(expected["b"], actual["b"], places=6)

  @jtu.skip_on_devices('tpu')
  def test_cg_matmul(self):
    A = CustomOperator(2 * jnp.eye(3))
    b = jnp.arange(9.0).reshape(3, 3)
    expected = b / 2
    actual, _ = jax.scipy.sparse.linalg.cg(A, b)
    self.assertAllClose(expected, actual)

  def test_cg_errors(self):
    A = lambda x: x
    b = jnp.zeros((2,))
    with self.assertRaisesRegex(
        ValueError, "x0 and b must have matching tree structure"):
      jax.scipy.sparse.linalg.cg(A, {'x': b}, {'y': b})
    with self.assertRaisesRegex(
        ValueError, "x0 and b must have matching shape"):
      jax.scipy.sparse.linalg.cg(A, b, b[:, np.newaxis])
    with self.assertRaisesRegex(ValueError, "must be a square matrix"):
      jax.scipy.sparse.linalg.cg(jnp.zeros((3, 2)), jnp.zeros((2,)))
    with self.assertRaisesRegex(
        TypeError, "linear operator must be either a function or ndarray"):
      jax.scipy.sparse.linalg.cg([[1]], jnp.zeros((1,)))

  def test_cg_without_pytree_equality(self):

    @register_pytree_node_class
    class MinimalPytree:
      def __init__(self, value):
        self.value = value
      def tree_flatten(self):
        return [self.value], None
      @classmethod
      def tree_unflatten(cls, aux_data, children):
        return cls(*children)

    A = lambda x: MinimalPytree(2 * x.value)
    b = MinimalPytree(jnp.arange(5.0))
    expected = b.value / 2
    actual, _ = jax.scipy.sparse.linalg.cg(A, b)
    self.assertAllClose(expected, actual.value)

  def test_cg_weak_types(self):
    x, _ = jax.scipy.sparse.linalg.bicgstab(lambda x: x, 1.0)
    self.assertTrue(dtypes.is_weakly_typed(x))

  # BICGSTAB
  @jtu.sample_product(
    shape=[(5, 5)],
    dtype=[np.float64, np.complex128],
    preconditioner=[None, 'identity', 'exact', 'random'],
  )
  def test_bicgstab_against_scipy(
      self, shape, dtype, preconditioner):
    if not config.jax_enable_x64:
      raise unittest.SkipTest("requires x64 mode")

    rng = jtu.rand_default(self.rng())
    A = rng(shape, dtype)
    b = rng(shape[:1], dtype)
    M = self._fetch_preconditioner(preconditioner, A, rng=rng)

    def args_maker():
      return A, b

    self._CheckAgainstNumpy(
        partial(scipy_bicgstab, M=M, maxiter=1),
        partial(lax_bicgstab, M=M, maxiter=1),
        args_maker,
        tol=1e-5)

    self._CheckAgainstNumpy(
        partial(scipy_bicgstab, M=M, maxiter=2),
        partial(lax_bicgstab, M=M, maxiter=2),
        args_maker,
        tol=1e-4)

    self._CheckAgainstNumpy(
        partial(scipy_bicgstab, M=M, maxiter=1),
        partial(lax_bicgstab, M=M, maxiter=1),
        args_maker,
        tol=1e-4)

    self._CheckAgainstNumpy(
        np.linalg.solve,
        partial(lax_bicgstab, M=M, atol=1e-6),
        args_maker,
        tol=1e-4)

  @jtu.sample_product(
    shape=[(2, 2), (7, 7)],
    dtype=float_types + complex_types,
    preconditioner=[None, 'identity', 'exact'],
  )
  @jtu.skip_on_devices("gpu")
  def test_bicgstab_on_identity_system(self, shape, dtype, preconditioner):
    A = jnp.eye(shape[1], dtype=dtype)
    solution = jnp.ones(shape[1], dtype=dtype)
    rng = jtu.rand_default(self.rng())
    M = self._fetch_preconditioner(preconditioner, A, rng=rng)
    b = matmul_high_precision(A, solution)
    tol = shape[0] * float(jnp.finfo(dtype).eps)
    x, info = jax.scipy.sparse.linalg.bicgstab(A, b, tol=tol, atol=tol,
                                               M=M)
    using_x64 = solution.dtype.kind in {np.float64, np.complex128}
    solution_tol = 1e-8 if using_x64 else 1e-4
    self.assertAllClose(x, solution, atol=solution_tol, rtol=solution_tol)

  @jtu.sample_product(
    shape=[(2, 2), (4, 4)],
    dtype=float_types + complex_types,
    preconditioner=[None, 'identity', 'exact'],
  )
  @jtu.skip_on_devices("gpu")
  def test_bicgstab_on_random_system(self, shape, dtype, preconditioner):
    rng = jtu.rand_default(self.rng())
    A = rng(shape, dtype)
    solution = rng(shape[1:], dtype)
    M = self._fetch_preconditioner(preconditioner, A, rng=rng)
    b = matmul_high_precision(A, solution)
    tol = shape[0] * float(jnp.finfo(A.dtype).eps)
    x, info = jax.scipy.sparse.linalg.bicgstab(A, b, tol=tol, atol=tol, M=M)
    using_x64 = solution.dtype.kind in {np.float64, np.complex128}
    solution_tol = 1e-8 if using_x64 else 1e-4
    self.assertAllClose(x, solution, atol=solution_tol, rtol=solution_tol)
    # solve = lambda A, b: jax.scipy.sparse.linalg.bicgstab(A, b)[0]
    # jtu.check_grads(solve, (A, b), order=1, rtol=3e-1)


  def test_bicgstab_pytree(self):
    A = lambda x: {"a": x["a"] + 0.5 * x["b"], "b": 0.5 * x["a"] + x["b"]}
    b = {"a": 1.0, "b": -4.0}
    expected = {"a": 4.0, "b": -6.0}
    actual, _ = jax.scipy.sparse.linalg.bicgstab(A, b)
    self.assertEqual(expected.keys(), actual.keys())
    self.assertAlmostEqual(expected["a"], actual["a"], places=5)
    self.assertAlmostEqual(expected["b"], actual["b"], places=5)

  def test_bicgstab_weak_types(self):
    x, _ = jax.scipy.sparse.linalg.bicgstab(lambda x: x, 1.0)
    self.assertTrue(dtypes.is_weakly_typed(x))

  @jtu.skip_on_devices('tpu')
  def test_bicgstab_matmul(self):
    A = CustomOperator(2 * jnp.eye(3))
    b = jnp.arange(9.0).reshape(3, 3)
    expected = b / 2
    actual, _ = jax.scipy.sparse.linalg.bicgstab(A, b)
    self.assertAllClose(expected, actual)

  # GMRES
  @jtu.sample_product(
    shape=[(3, 3)],
    dtype=[np.float64, np.complex128],
    preconditioner=[None, 'identity', 'exact', 'random'],
    solve_method=['incremental', 'batched'],
  )
  def test_gmres_against_scipy(
      self, shape, dtype, preconditioner, solve_method):
    if not config.x64_enabled:
      raise unittest.SkipTest("requires x64 mode")

    rng = jtu.rand_default(self.rng())
    A = rng(shape, dtype)
    b = rng(shape[:1], dtype)
    M = self._fetch_preconditioner(preconditioner, A, rng=rng)

    def args_maker():
      return A, b

    self._CheckAgainstNumpy(
        partial(scipy_gmres, M=M, restart=1, maxiter=1),
        partial(lax_gmres, M=M, restart=1, maxiter=1, solve_method=solve_method),
        args_maker,
        tol=1e-10)

    self._CheckAgainstNumpy(
        partial(scipy_gmres, M=M, restart=1, maxiter=2),
        partial(lax_gmres, M=M, restart=1, maxiter=2, solve_method=solve_method),
        args_maker,
        tol=1e-10)

    self._CheckAgainstNumpy(
        partial(scipy_gmres, M=M, restart=2, maxiter=1),
        partial(lax_gmres, M=M, restart=2, maxiter=1, solve_method=solve_method),
        args_maker,
        tol=1e-9)

    self._CheckAgainstNumpy(
        np.linalg.solve,
        partial(lax_gmres, M=M, atol=1e-6, solve_method=solve_method),
        args_maker,
        tol=1e-8)

  @jtu.sample_product(
    shape=[(2, 2), (7, 7)],
    dtype=float_types + complex_types,
    preconditioner=[None, 'identity', 'exact'],
    solve_method=['batched', 'incremental'],
  )
  @jtu.skip_on_devices("gpu")
  def test_gmres_on_identity_system(self, shape, dtype, preconditioner,
                                    solve_method):
    A = jnp.eye(shape[1], dtype=dtype)

    solution = jnp.ones(shape[1], dtype=dtype)
    rng = jtu.rand_default(self.rng())
    M = self._fetch_preconditioner(preconditioner, A, rng=rng)
    b = matmul_high_precision(A, solution)
    restart = shape[-1]
    tol = shape[0] * float(jnp.finfo(dtype).eps)
    x, info = jax.scipy.sparse.linalg.gmres(A, b, tol=tol, atol=tol,
                                            restart=restart,
                                            M=M, solve_method=solve_method)
    using_x64 = solution.dtype.kind in {np.float64, np.complex128}
    solution_tol = 1e-8 if using_x64 else 1e-4
    self.assertAllClose(x, solution, atol=solution_tol, rtol=solution_tol)

  @jtu.sample_product(
    shape=[(2, 2), (4, 4)],
    dtype=float_types + complex_types,
    preconditioner=[None, 'identity', 'exact'],
    solve_method=['incremental', 'batched'],
  )
  @jtu.skip_on_devices("gpu")
  def test_gmres_on_random_system(self, shape, dtype, preconditioner,
                                  solve_method):
    rng = jtu.rand_default(self.rng())
    A = rng(shape, dtype)

    solution = rng(shape[1:], dtype)
    M = self._fetch_preconditioner(preconditioner, A, rng=rng)
    b = matmul_high_precision(A, solution)
    restart = shape[-1]
    tol = shape[0] * float(jnp.finfo(A.dtype).eps)
    x, info = jax.scipy.sparse.linalg.gmres(A, b, tol=tol, atol=tol,
                                            restart=restart,
                                            M=M, solve_method=solve_method)
    using_x64 = solution.dtype.kind in {np.float64, np.complex128}
    solution_tol = 1e-8 if using_x64 else 1e-4
    self.assertAllClose(x, solution, atol=solution_tol, rtol=solution_tol)
    # solve = lambda A, b: jax.scipy.sparse.linalg.gmres(A, b)[0]
    # jtu.check_grads(solve, (A, b), order=1, rtol=2e-1)

  def test_gmres_pytree(self):
    A = lambda x: {"a": x["a"] + 0.5 * x["b"], "b": 0.5 * x["a"] + x["b"]}
    b = {"a": 1.0, "b": -4.0}
    expected = {"a": 4.0, "b": -6.0}
    actual, _ = jax.scipy.sparse.linalg.gmres(A, b)
    self.assertEqual(expected.keys(), actual.keys())
    self.assertAlmostEqual(expected["a"], actual["a"], places=5)
    self.assertAlmostEqual(expected["b"], actual["b"], places=5)

  @jtu.skip_on_devices('tpu')
  def test_gmres_matmul(self):
    A = CustomOperator(2 * jnp.eye(3))
    b = jnp.arange(9.0).reshape(3, 3)
    expected = b / 2
    actual, _ = jax.scipy.sparse.linalg.gmres(A, b)
    self.assertAllClose(expected, actual)

  @jtu.sample_product(
    shape=[(2, 2), (3, 3)],
    dtype=float_types + complex_types,
    preconditioner=[None, 'identity'],
  )
  def test_gmres_arnoldi_step(self, shape, dtype, preconditioner):
    """
    The Arnoldi decomposition within GMRES is correct.
    """
    if not config.x64_enabled:
      raise unittest.SkipTest("requires x64 mode")

    rng = jtu.rand_default(self.rng())
    A = rng(shape, dtype)
    M = self._fetch_preconditioner(preconditioner, A, rng=rng)
    if preconditioner is None:
      M = lambda x: x
    else:
      M = partial(matmul_high_precision, M)
    n = shape[0]
    x0 = rng(shape[:1], dtype)
    Q = np.zeros((n, n + 1), dtype=dtype)
    Q[:, 0] = x0 / jnp.linalg.norm(x0).astype(dtype)
    Q = jnp.array(Q)
    H = jnp.eye(n, n + 1, dtype=dtype)

    @jax.tree_util.Partial
    def A_mv(x):
      return matmul_high_precision(A, x)
    for k in range(n):
      Q, H, _ = sp_linalg._kth_arnoldi_iteration(
          k, A_mv, M, Q, H)
    QA = matmul_high_precision(Q[:, :n].conj().T, A)
    QAQ = matmul_high_precision(QA, Q[:, :n])
    self.assertAllClose(QAQ, H.T[:n, :], rtol=1e-5, atol=1e-5)

  def test_gmres_weak_types(self):
    x, _ = jax.scipy.sparse.linalg.gmres(lambda x: x, 1.0)
    self.assertTrue(dtypes.is_weakly_typed(x))


if __name__ == "__main__":
  absltest.main(testLoader=jtu.JaxTestLoader())